论文标题
需要对某些退化椭圆运算符的不细胞的对数估算值的必要性
Necessity of a logarithmic estimate for hypoellipticity of some degenerately elliptic operators
论文作者
论文摘要
本文扩展了一类退化的椭圆运算符，其低细胞性不仅需要在各个方向上衍生物的衍生物的对数增益。 Hoshiro和Morimoto在80年代后期的作品表征了衍生物具有超级同源化的必要性，以使退化运算符和一些非脱位操作员（如Laplacian）的多层次达成。我们认为的操作员是相似的，但更一般。我们检查了$ l_1（x）+g（x）l_2（y）$的运算符，其中$ l_1（x）$是一维，$ g（x）$可能会消失。本文的论点基于光谱投影，对频谱微分方程的分析以及标准和操作员适应的衍生物之间的插值。与文献中的先前结果不同，我们的结果不需要在非分类方向上明确的分析结构。实际上，我们的结果允许非分析部分的非分析甚至非平滑系数。
This paper extends a class of degenerate elliptic operators for which hypoellipticity requires more than a logarithmic gain of derivatives of a solution in every direction. Work of Hoshiro and Morimoto in late 80s characterized a necessity of a super-logarithmic gain of derivatives for hypoellipticity of a sum of a degenerate operator and some non-degenerate operators like Laplacian. The operators we consider are similar, but more general. We examine operators of the form $L_1(x)+g(x)L_2(y)$, where $L_1(x)$ is one-dimensional and $g(x)$ may itself vanish. The argument of the paper is based on spectral projections, analysis of a spectral differential equation and interpolation between standard and operator-adapted derivatives. Unlike prior results in the literature, our results do not require explicit analytic construction in the non-degenerate direction. In fact, our result allows non-analytic and even non-smooth coefficients for the non-degenerate part.
